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Advanced Math / Nonlinear functions Difficulty: Hard

$f (x) = |59 - 2 x|$

The function $f$ is defined by the given equation. For which of the following values of $k$ does $f left parenthesis k right parenthesis equals 3 k$ ?

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Explanation

Choice A is correct. The value of $k$ for which $f left parenthesis k right parenthesis equals 3 k$ can be found by substituting $k$ for $x$ and $3 k$ for $f (x)$ in the given equation, $f (x) = |59 - 2 x|$ , which yields $3 k = |59 - 2 k|$ . For this equation to be true, either $- 3 k = 59 - 2 k$ or $3 k = 59 - 2 k$ . Adding $2 k$ to both sides of the equation $- 3 k = 59 - 2 k$ yields $- k = 59$ . Dividing both sides of this equation by $negative 1$ yields $k = - 59$ . To check whether $negative 59$ is the value of $k$ , substituting $negative 59$ for $k$ in the equation $3 k = |59 - 2 k|$ yields $3 (- 59) = |59 - 2 (- 59)|$ , which is equivalent to $- 177 = |177|$ , or $- 177 = 177$ , which isn't a true statement. Therefore, $negative 59$ isn't the value of $k$ . Adding $2 k$ to both sides of the equation $3 k = 59 - 2 k$ yields $5 k equals 59$ . Dividing both sides of this equation by $5$ yields $k = \frac{59}{5}$ . To check whether $StartFraction 59 Over 5 EndFraction$ is the value of $k$ , substituting $StartFraction 59 Over 5 EndFraction$ for $k$ in the equation $3 k = |59 - 2 k|$ yields $3 (\frac{59}{5}) = |59 - 2 (\frac{59}{5})|$ , which is equivalent to $\frac{177}{5} = |\frac{177}{5}|$ , or $\frac{177}{5} = \frac{177}{5}$ , which is a true statement. Therefore, the value of $k$ for which $f left parenthesis k right parenthesis equals 3 k$ is $StartFraction 59 Over 5 EndFraction$ .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.